The arxiv version of the paper is here. Since I’m quoted in the Science News piece saying something to the extent that I have my reservations but think it’s a promising direction of study, I have gotten a lot of questions about negative masses in General Relativity lately. So here a clarification.

First one has to be careful what one means with mass. There are three types of masses: inertial mass, passive gravitational mass, and active gravitational mass. In General Relativity these masses, or their generalization in terms of tensors respectively, are normally assumed to be identical.

The equality of inertial and passive gravitational mass is basically the equivalence principle. The active gravitational mass is what causes space-time to bend; the passive gravitational mass is what couples to the space-time and determines the motion of particles in that background. The active and passive gravitational masses are identical in almost all theories I know. (The Schrödinger-Newton approach is the only exception that comes to mind). I doubt it is consistent to have them not be equal, but I am not aware of a proof for this. (I tried in the Schrödinger-Newton case, but it’s not as trivial as it looks at first sight.)

In General Relativity one further has to distinguish between the local quantities like energy-density and pressure and so on that are functions of the coordinates, and global quantities that describe the space-time at large. The total mass or energy in some asymptotic limit are essentially integrals over the local quantities, and there are several slightly different ways to define them.

The positive mass theorem, in contrast to what its name suggests, does not state that one cannot have particles with negative masses. It states instead, roughly, that if your local matter is normal matter and obeys certain plausible assumptions, then the total energy and mass are also positive. You thus cannot have stars with negative masses, regardless of how you bend your space-time. This isn’t as trivial a statement as it sounds because the gravitational interaction contributes to the definition of these integrated quantities. In any case, the positive mass theorem holds in space that is asymptotically flat.

Now what they point out in the new paper is that for all we know we don’t live in asymptotically flat space, but we live in asymptotic de-Sitter space because observational evidence speaks for a positive cosmological constant. In this case the positive mass theorem doesn’t apply. Then they go on to construct a negative mass solution in asymptotic de Sitter space. I didn’t check the calculation in detail, part of it is numerical, but it all sounds plausible to me.

However, it is somewhat misleading to call the solution that they find a negative mass solution. The cosmological constant makes a contribution to the effective mass term in what you can plausibly interpret as the gravitational potential. Taken together both, the effective mass in the potential is positive in the region where this solution applies. The local mass (density) is also positive by assumption. (You see this most easily by looking at fig 1 in the paper.)

Selling this as a negative mass solution is like one of these ads that say you’ll save 10$ if you spend at least $100 – in the end your expenses are always positive. The negative mass in their solution corresponds to the supposed savings that you make. You never really get to see them. What really matters are the total expenses. And these are always positive. There are thus no negative mass particles in this scenario whatsoever. Further, the cosmological constant is necessary for these solutions to exist, so you cannot employ them to replace the cosmological constant.

It also must be added that showing the existence of a certain solution to Einstein’s field equations is one thing, showing that they have a reasonable chance to actually be realized in Nature is an entirely different thing. For this you have to come up with a mechanism to create them and you also have to show that they are stable. Neither point is addressed in the paper.

Advertisement break: If you want to know how one really introduces negative masses into GR, read this.

In the Science News article Andrew Grant quotes one of the authors as saying:

“Paranjape wants to look into the possibility that the very early universe contained a plasma of particles with both positive and negative mass. It would be a very strange cosmic soup, he says, because positive mass gravitationally attracts everything and negative mass repels everything.”

This is wrong. Gravitation is a spin-2 interaction. It is straightforward to see that this means that like charges attract and unlike charges repel. The charge of gravity is the mass. This does not mean that negative gravitational mass repels everything. Negative gravitational mass repels positive mass but attracts negative mass. If this wasn’t so, then you’d run into the above mentioned inconsistencies. The reason this isn’t so in the case considered in the paper is that they don’t have negative masses to begin with. They have certain solutions that basically have a gravitational attraction which is smaller than expected.

In summary, I think it’s an interesting work, but so far it’s an entirely theoretical construct and its relevance for the description of cosmological dynamics is entirely unclear. There are no negative mass particles in this paper in any sensible interpretation of this term.

A light beam and two mirrors in "flat" vacuum form a closed triangle. Its three interior angles sum to exactly 180°. Add gravitation. Said sum increases up to 540°. Decrease the 180° sum for anti-gravitation.

In kindergarten I failed.The vectors of light for Minkowski space simply failed to fullfill the definition given for zero vectors. And those light vectors were depicted graphically as 'cones'as well ...along with their mislabeling.

You are never daunted. With your "call it whatever you like" you went on and moved forward.

I didn't move at all and threw a temper tantrum. Seeking comfort I turned to ice cream cones instead of light cones.

Giotis: A smart question, as always... Again it depends on which mass you are referring to. If you mean the inertial mass, you replace m->-m in the Lagrangian, which brings with it all sorts of problems. If you mean the gravitational mass, it's not the mass itself that you have to change but the coupling of the field to gravity. If you want to bring in a difference between the active and the passive gravitational mass you have to figure out a way to get a different coupling in the field equation than in the geodesic equation. Since one can derive the latter from the former, I don't think this is consistently possible. (The reason this seems to be working in the Schroedinger-Newton approach is that the expectation values are still the same.) Best,

Uncle AI,I have trouble or do not understand your description or how it fits in the chemistry or physics here or elsewhere.In Euclid the distance between two equilateral triangles meeting at a point on a base line of the same area cannot be proven equal to a side of either at their tops.Yes there are true and plausible statements here on the non Euclidean.I could not link to your picture. Right Hand Rule.String or other models aside, why can it not be 720 degrees as 8 right angles?for full circle and something negative or hidden? Enter abstract multiples of 27, three cubed twists expressed linearly? 8x27 = 6 cubed. How many steps then? 5 material and 3 hidden or better yet 10 at rest and 6 electromagnetic in the 4x4 matrix?

Nice, Bee. I think it's a little misleading, though, to talk about three kinds of mass -- especially since you immediately qualify the statement to note that these are identical to relativists using continuous function models.

Negative mass has been known as a element of special relativity from the beginning, as a consequence of choosing a particle having mass with zero momentum -- implied in the unreduced equation E^2 = m^2c^4 + (pc)^2 -- p being momentum.

The question to ask is not whether negative mass exists; it clearly does, as an artifact of continuous function physics. Einstein was always careful, however, to distinguish physically real solutions from computational artifacts. His definition that makes spacetime physically real is what makes Mbarek's and Paranjape's idea plausible: Einstein averred that a physically real spacetime is " ... independent in its properties, having a physical effect though not itself influenced by physical conditions." (*Meaning of Relativity* 1956 Princeton Press)

So to cast spacetime in terms of a perfect fluid is exactly the right model for general relativity cosmology -- negative mass has a causal role only absent momentum. As the authors conclude: " ... in the early universe there would be a plasma of positive and negative mass particles. Such a plasma would in principle cause an ef- fective screening of gravitational waves, being essentially opaque for frequencies below the plasma frequency."

So once momentum is introduced, positive mass is the only solution of mass-energy equivalence. I think this model will get attention from cosmologists, because it potentially answers deep and important questions, such as that of Lemaitre, who asked if the universe could ever have been at rest. (at rest relative to what?)

Einstein refined and changed our sense of the meaning of what is mass. The scientific method while in the main is stable incorporating earlier models to a more general picture understands all models as provisional. Einstein proposed an even higher steady state model of the universe so as part of the picture such models are doubly provisional said driven by ignorance to explore objectively.QM theory said to be a measure of our ignorance would be doubly ignorant and ideas like inflation or cosmological constants, Casmir measure and so on are doubly provisional.

Bee,what do you think of Luc Blanchet's work where he too has proposed mass dipolar fluid e (positive mass and negative mass separated by distance if I understand correctly). It also uses indirectly concept of negative mass.

Placing a probe in the path of an electromagnetic wave train one sees that the electric and magnetic vectors are perpendicular to one another, and to the direction of the wave's propagation, and are always in phase with one another. The electric vector is periodically alternating between positive and negative, and likewise, the magnetic vector is periodically alternating between north and south polarity.

Now replace the electric vector with a 'time' vector, and the magnetic vector with a 'length' vector. As this hypothetical wave propagates the time vector alternates between forward and reverse time, while the length vector alternates between positive and negative length.

During the interval when time is reversed and length goes negative, it would correspond to a negative energy state of the vacuum. During the excursions into forward moving time and positive length values, it would constitute a positive state of the vacuum.

But all lengths are positive in our universe, and time always moves in the forward direction, making such a length-time wave impossible if it were imbedded 'only' in our universe. But suppose our universe is bounded by a higher dimensional space, such as envisioned in various extra dimensional models, in which time and length are effectively reversed.

If the fundamental particles of our universe were stuck at the boundary of this higher dimensional space, as per Randall-Sundrum 1/2, and other models our electrons, quarks, etc, could experience such waves, and thus be subject to positive and negative vacuum excursions from their point of view.

Thanks for the post, but I have a very minor English nitpick, concerning:

"Since I’m quoted in the Science News piece saying something to the extent that ..."

I have seen the idiom "saying something to the effect that ..." but not "to the extent that ...". The former means that you are about to give an interpretation which has the same meaning (effect) as some source material without using the same words as the source. So I think using "effect" instead of "extent" would be a small improvement.

(Previously I have not offered such small English corrections when the meaning was obvious, but I noticed that when others do you don't seem annoyed - so far.)

Thanks for this. No, I'm not annoyed. I spend a big part of my days correcting my daughters grammar - how else are they supposed to learn it? Thinking about it, I am not actually sure where I might have gotten that expression from. It's quite possible that I heard it somewhere and just misheard it, I will keep it in mind. Best,

Hi Sabine,Is this comment of you "Negative gravitational mass repels positive mass but attracts negative mass." true? I thought that a negative mass source produces repulsive geodesics around itself along which all masses are repelled irrespective of whether they are positive or negative. Correct me if I'm wrong. Thanks.

sppondee: You are wrong. There is no such thing as a 'negative geodesic'. Geodesics are geodesics, and according to the equivalence principle they are independent of the mass of the falling object. You can easily convince yourself that what I said is correct, it's a very basic calculation about the coupling of the currents in the interaction terms, you find it eg in the Feynman lectures. If the force carrier is a spin two field, like charges attract, unlike charges repel. For a spin-1 field it's the opposite - the sign depends on the spin. Best,

Hi B, thanks for your prompt response.I didn't say "negative geodesic". Maybe you mean "repulsive geodesic". I was thinking that if a mass is negative it would make affine connections with different sign than for the case of a positive mass. Wouldn't it then imply all the masses are repulsive sources of gravity? If negative attracts negative, why isn't a negative mass star possible? Thanks.

Sspondee: You can't just swap the sign in the affine connection, that's not a generally covariant prescription. What you can do is introduce a second covariant derivative, which is torsion-free but non-metric, and if you want to know how that works, please read the paper I mentioned in the above post (und Self-advertisement). Yes, negative mass stars are possible. Which, if you think about it, tells you how nonsensical the quoted comment from the author above is because if it was correct what he said then their paper would be wrong. Best,

The negative mass is not a new concept in theoretical physics at all, it exists here in concepts of so-called gravitomagnetism (which supports the negative gravitational charge) and mirror matter. The dark matter exhibits many characteristics of matter of negative gravitational charge. We even have observational evidence for it in lensing of free space between galaxies. The basic idea here is, the negative mass is repelled from observable matter, so it gets concentrated at the free space between massive bodies.

Don a pair of polarized 3-D movie glasses. Look in a mirror, then close one eye. What is the sight of one eye not looking? It works for the other eye, too. What happens if you close both eyes? You see each other.

Uncle AI,You too address the heart of the matter.Do two bits of antimatter repel?Sabine mentioned Affine geometry and I looked it up. If we apply the concept to the Planck background we see now from the great observation of nature that it makes it hard to sort out the macro and micro cosm. Doubly defining this Flatland abstraction not only are the signs independent of issues of measure between vector points but not which eye we close to analyze the data either.

Furthermore, Uncle AI, (and Zephir talking of paradoxes finally on a fundamental level) I have looked with.both eyes closed. I have more to say about this as science fiction but QM double slit and dark matter metaphors are not fantasy - more like philosophy where it meets science methods.So looking with both eyed closed I have seen pictures from others with intent to share if both sides of our brain is focused and we both are objective to not guess say s colored number at our point of uncertainty and prejudice.But yours is a good arguement. All I can say is try it.

Hi Bee,I'm not understanding some of your comments regarding repulsion. Here are my thoughts, and can you tell me which you agree with (or where I go wrong)?

1] If you take a classical action and multiply it by a constant, the same equations of motion result.

2] The action for a test particle in a curved background is just S = - \int m d\tau, where m is the mass and \tau is the path length of its worldline. We find that particles travel on geodesics.

3] From 1 and 2, multiplying the particle's mass by (-1) changes nothing.

4] Restating #3 (again, from 1 and 2): The path a negative mass test particle takes in a Schwarzchild background (outside some spherical static positive mass), is the same as the path a positive mass test particle would take with the same initial conditions.

It makes no sense to me that you say "There is no such thing as a 'negative geodesic'. Geodesics are geodesics, and according to the equivalence principle they are independent of the mass of the falling object." but then immediately claim a negative mass repels a positive mass. Don't you see the contradiction!?

You are saying the same thing I said above. I don't know why you are trying to construct this as a disagreement. My comment "There is no such thing as a 'negative geodesic'." means exactly this: a geodesic is a geodesic, and changing the INERTIAL mass of the particle doesn't do anything about this.

I also said above that the only way to introduce negative masses consistently into GR (meaning: generally covariant) is to introduce a second covariant derivate (dropping the assumption of metricity). Then there are two types of paralell transport, ergo two types of geodesics. All of this btw is explained in my paper, which I recommend you read before you attempt further disagreements. Best,

Is there in some sense a negative probability? Should philosophers ask why the universe exists rather than why it does not, or ask what is the probability it exists?Surely an array of cards can have two different probabilities which amounts to a paradox that by shifting rows can pinpoint any card in the array. This is the power of squared Flatland representation.What is a contradiction but al zero's in a predicate logic, zero the impossible while one the possible all in the logic is a tautology? Is nature logical in this sense? Can it have three things taken distinctly four at a time? Given a null set it can be thought of as minus one which includes but does not explain the complex numbers. How many Higgs particles can be distinct from but one of them at a time turning binomial coefficients on end in Pascal's triangle.?Topology or discrete integers in number theory cannot explain unity alone over a QM model something both or either a zero or one. 2D or two geodesic paths are sufficient as there are no gray groups as is psi square or the extension to fuzzy logic. It occurred to me these considerations explain the terrain of Riemann 's real zeta zeros of a complex number. Others have mentioned the possibility too.I do not know how good these thoughts are but as it stand one could conclude the two physics cannot logically cotrespond. Enter interpretations as consciousness and one soul's view is to the other soul a stance as an unresolvable science fiction. Newton extended Pascal's triangle to negative values and went from that to invent the calculus.

Later in the comments you again say:"like charges attract, unlike charges repel."

I don't understand how you can claim you are saying the same thing I said, when I directly contradicted that with my statement #5.

I'm not trying to create disagreements. I'm trying to learn, and I'm trying to understand.

The predictions of GR seem unambiguous here: in GR a positive mass attracts both positive mass particles and negative mass particles. You seem to agree with this prediction, while also maintaining the contradictory position that opposite signed masses repel.

Can you at least understand why I am confused here? Please clarify and help me understand what you mean by this in your blog post.

I have explained this in my paper, which you find here. A positive gravitational mass attracts both negative and positive INERTIAL mass, because the geodesics don't depend on the inertial mass to begin with. But negative inertial mass is a flawed concept anyway. A positive gravitational mass attracts positive GRAVITATIONAL mass but repels negative GRAVITATIONAL mass (the inertial energies remain positive). The only way to consistently introduce negative gravitational masses is by using a second covariant derivative, as I now say for the third time, and as you can read in the paper which I have now mentioned for the third time.

I have quoted the statement from the author in my post to explain why it's wrong, so why are you surprised that I am disagreeing with him?

Hey Bee, I'm very cautious since you seem to be quite irritated by the repeated questions by me(spondee) and another commenter(Student4life). If you are, I'm sorry. I'm just curious. I read your paper and Bondi's paper mentioned there. You say, Bondi pointed out a contradiction, but I see, Bondi explaining a funny situation with a pair of positive- and negative-mass particles within the context of GR. A positive-mass particle is attracting everything including a negative-mass particle and likewise a negative-mass particle repelling everything including a positive-mass particle. Then the situation is, a pair of particles accelerating spontaneously to one direction. Funny and not happening in our Universe. But I don't see any inconsistencies you are referring to. The phenomenon belongs to another universe, but still viable. Advocates of multiverse idea might think it's okay. Bi-metric theory looks more esoteric to me. I would rather think of a mechanism which would separate positive-mass particles and negative-mass particles very far from each other. Hope I didn't disturb your inner peace.

Thanks.

PS. Your previous comment "If the force carrier is a spin two field, like charges attract, unlike charges repel." is true if you do QFT in a flat spacetime. But we are here considering two particles moving in a space-time warped by each other. Is the conclusion still valid?

Sppondee , You seem to be at inner peace with your references to logic terms. Is it not possible Sabrine has both eyes open on this issue and you and your friend are only half conscious in understanding?

Now, a topological problem such as connecting three objects to three other objects is a torus. One can see it as the ferryman taking the fox, chicken, and corn across a river that any two of them are not eaten.

Computers, presumably based on logic have a hard time with such a problem.Think of the Ferryman as wiser taking actions from a wider space. The problem being the computer cannot see there are two solutions. What is so hard about seeing they both arrive at the same positive place? Sometimes things are symmetric and conserved only if all thtee as in CPT are considered together.

I'm not irritated, I'm annoyed because I don't like to repeat myself. Which you'd know if you'd read my previous post.

In any case, the statement that I made is true not only in flat space but in any fixed background, but this isn't the point. The point is it is correct in some limit and whatever you do, you will have to reproduce this limit. If you believe you can have a negative gravitational mass that 'repels everything' you will never get this limit, ergo your theory is wrong, or maybe not even wrong.

About Bondi - well, he arrives at self-accelerating solutions. That is inconsistent enough for me. See, the contradiction that he has is exactly because the limit that you are asking for does *not* work. You can derive the geodesic equation from the field equation, so if you have two active grav masses that supposedly repel each other, then you shouldn't get the geodesic equation - except that Bondi didn't know any other equation that could possibly come out. This is what you get when you assume both metricity and torsion-free-ness for the connection, which he has implicitly done. Drop metricity and you can do it just fine. All of this is of course explained in my paper...

Uncle AI Between the micro and macro scales the tension reverses as with DNA. The experiment from a higher space and not the mutual implication nature discerns as useful chiraility as three space effects is balanced so observed torsion would be zero as in Sabine's paper describing two such fields and two sufficient to the level of description.One or both at half intensity would fluoresce, or gain unobservable mass that breaks apart the crystals at a certain threshold.Why 11? How many ways can you unfold a cube and can any of them tile a plane? Would photons replicating cycled thru a wormhole cause its collaspe? Would one photon set off many cells in photosynthesis seemingly at a distance? Where are the monopoles ad hoc in Maxwell's equations? In these quasi-discrete considerations how accurately can we draw black holes between two spaces, points still or lines dynamic that a unit cell is invariant in form and the information balanced?

The situation concerning the definition of mass in General Relativity is rather confusing. Einstein certainly realized this but for historical reasons did not have enough experimental wisdom to solve the problem.

The fundamental reason to problems is of course that bringing matter to empty Minkowski space breaks its symmetries and leads to the loss of Noether theorem, which would give four-momentum and angular momentum as integrals and allow also quantum mechanical treatment of conservation laws as a consequence of symmetries. This loss of symmetry is very probably also behind the problems in quantization of general relativity: successful quantum theoryis mostly representation theory for symmetries!

One cay try to identify mass as asymptotic behaviour of solution assuming asymptotic flatness. This is ok for simple stationary solutions with spherical symmetry. The pseudo tensor constructions are to my opinion hopelessly ad hoc.

Twistor enthusiast could tell also about another problem: only empty Minkowski space allows twistor space. Twistors have shown their power so that there is a real need to get twistors but general relativity does not allow them.

Both problems find a beautiful solution when space-times are regarded as 4-surfaces in imbedding space M^4xS, S some compact Euclidian space. If also S is required to possess twistor space with Kaehler structure, only one option remains: S=CP_2 yielding explanation for standard model symmetries and allowing a geometrization of electroweak and color gauge fields. Poincare symmetries are lifted to the level of imbedding space. Conservation laws are exact and one gets rid of the problems with mass.

GRT space-time follows from this picture as an approximate macroscopic concept. In long length scales the space-time as surfaces is many-sheeted in the sense that there are several space-time sheets having non-empty 4-D projection to a given region of M^4. GRT space-time is obtained by replacing these sheets with a region of M^4 endowed with a metric, which is sum of Minkowski metric and deviations of the metrices of sheets from Minkowski metric. Gauge potentials are defined in analogous manner in terms of induced gauge potentials.

This allows to circumvent the objection that the reduction of all field variables to four CP_2 coordinates and their gradients - although it is a huge local simplification - cannot make sense since one would lose linear superposition. Linear superposition of fields is replaced with that for their effects: small space-time surface representing test particle touches the spacetime sheets of many-sheeted spacetime and experiences the sum of their effects.

So it goes, confusing the map with the terrain. Only there are at least two maps "bimetric" and two terrains. Only hints of five and greater as if unity in the physics. No twisters no gravity, no gravity no mass. And we have not as artist or half blind mapmakers looked into the one way mirror of the grid itself. Positive bubbles in infinitely negative space? These spherical Tori can come in distinct or illustions of numbers and topology. When scale and measure is everything so is embedding if the question of origins and endings beyond vectors and variance is not so vanishingly far away.

Now, Otto, you put on your thinking cap and tell us how Te's 3 atoms/unit cell can be a chiral geometry in a periodic crystal lattice. Chiral objects in 3-D must consist of at least four points that are not colinear, not coplanar, and have no S_n improper symmetry axes.

I am not sure of the terminology if any and I cannot follow your links as your consensus or documentation on this phone. But the question is deep and right in topic.

1 ) chirality emerges from three spaces mapped in two directions into each other so is not as fundamental as let us say ordered point like objects.

2) nuclear ressonance in pairs as projection are five or so times greater.

3) space A is flat yet multidimensional. Space B and C are represented as 3D volume.

4) between B of a recursive pattern as if spheres embedded in smaller spheres and so on and C ordered as sheets along the lines of Marti PitkanenFrom B as linear C and A break the continuity of the pattern.

5) from B or C into A circles and lines are interchanged in any case.

6) the broken pattern in A or C is preserved in form but A has an opposite sign of C at beginning and end of ordering as two or four things, 4786 L into 6784 R.

QED and well, QED :-)

But I do not have a Nominalism fot thisthat is more than s thought experiment but I do call it Triniquad or Triniquart although this Trinique is not a very poetic name.

I awoke in the wee hours last night due to reactivation of an intensely painful back injury (herniated disc), and had sort of a revelation about negative mass. Not totally sure these ideas are sensible, as it was hard to concentrate with pain.

Imagine the existence of a 'mirror' universe where length scales are negative and the arrow of time flows backwards, from our perspective. The matter in such a universe, from our perspective, would occupy negative volume. But to a sentient being composed of this matter, and residing in this mirror universe, the laws of Special and General Relativity would be indistinguishable from what we observe in our universe. From our perspective, all the gravitational fields, and matter that creates them, would possess negative energy. But, conversely, a being in this mirror universe would perceive all our matter and gravitational fields to have a negative energy state.

Taking this gedankenexperiment a little further, imagine an intrepid astronaut, in this mirror universe, entering a region of gravitational field so strong, that his length scales are cut in half, and time, correspondingly, flows at half its usual rate, from the perspective of this astronaut’s buddy in another ship far from this gravitational field. But from our perspective his lengths would appear to be getting longer, since they are negative to begin with. For instance a negative 2 meter rod would now become a negative 1 meter rod, so from our viewpoint would it is 1 meter longer. Likewise, with the flow rate of time. This astronaut’s buddy would note that his clocks were running half as fast as his own ship’s clocks, but we would see the reverse - a doubling of the clock rate aboard this astronaut’s ship that is exploring the intense gravitational field.

Now these observations lead to something really interesting. Our Universe has vast voids where there is apparently no normal matter. If negative matter is in these regions, and causing these matter gaps it is not visible via the electromagnetic spectrum. But if negative matter/energy is sequestered only on the above hypothesized mirror universe (3+1 brane), in accord with the various extra-dimension theories, presumably only gravitons from this mirror universe can cross over into our own universe. These gravitons, having a negative energy state, would communicate a repulsive gravitational force into our universe, but would be completely invisible, except for their effect.

Interesting, five fold shadows projected down do not represent five space fillers in normal 3D. In this truncated octahedron the hex faces would be slightly concave (negative) . So there is a flow as if a Superconductive electron free of any atom. Lord Kelvin and Ferdov's solid abstractly and globally conflict here.The outside and the inside of this half global 4D implied vortex even with inversion are missing (a unit cell of 8 quarter atoms). Now solve for nucleon pairs such asymmetry, how many of the 1152 or so are achiral?SPD = 2,4,10. But I do not understand your point 2) . In the Bizzaro world of super(man)symmetry theory cannot make red kryptonite unless we define initial or stable points on the six half atom square flat faces forever far from their main diagonal. Thanks for a little presumably empirical confirmation of my mathematical recreations. :-)

Uncle AII've known about this distorted space filler for quite some time. I posted it on Facebook recently before you asked but could not tag you. It can be seen as 24 dimensional lattices discretely but also the monster symmetry 240 lattice. Here fundamental physics asks some of our deepest questions on what is real or fantasy in questions of what is negative in more general spacetime. It is not everything only the Euclidean stance.

A negative mass matter in action during collisions of galaxies at distance. The negative mass matter is repulsed mutually, but still attracted to normal matter and it forms an invisible layer of dark matter around older and heavier galaxy, which prohibits their collisions. The so-called crackpots represent a "negative mass" society: they're repelled with mainstream and they're even fighting each other. They're still attracted to mainstream forums and discussions for to promote their ideas.

How the negative mass in dark matter works? It exhibits an excess of longitudinal/scalar waves of vacuum (virtual or sterile neutrinos) into account of transverse ones (virtual photons). The excess of radiative pressure keeps the dark matter particles at distance. In classical general relativity the so-called worm and white holes holes are also areas of negative mass and energy.

I think debates on theoretical models or playable games amount to whatever level one is comfortable or satisfied with. Not who is more wrong in what amounts to the higher game between two or more sentient human players or how we relate in love and war to our idea of the board or pieces given as resources objectively and mechanically.

Is such a vague parameter itself a game? Let us not forget this question if the human element, one that perhaps seeks a deeper level of anthropic implications even for machines.

My mathemathical recreations begin with design of logically extended games of n-dimensional chess but as a human player playing blind against the unknown there is only small temporary advantages knowing wider principles and structures. In short, this does not make us better chess players, or make us more intelligent in the longer run before some chosen stance as a draw crossing bridges to equilibrium.

I apollogize for the extended tangents to this blog that mercifully fades into time 's archive along with human errors. Yet the community of new topic posts follow in plausible order I take as a sign of higher intelligence and healthy adapting in our inquirey.

Two fingers inbound on a rigid rod can allow gravity to find the center yet if we are to build a dam we should begin in the center of water flow.

Where I have not commented on some of you, ideas from here today, I have passed thru such epiphanies long ago or missed some along the way being in the periphery.

You have shared the light of your stars Earthlings, Good inquiring to all of you.

As our gracious host ends many comments with important deeper meaning to you I bid you-all

Sorry to drop in so late, but I got here from a FB post. My main objection to negative mass has always been: it doesn't seem possible to convert it into the proper equivalent of "negative energy." Seriously, the other problems are weird or bad too, but imagine trying to convert a lump of negative mass into "negative-energy light rays." There is no way for Maxwell equations to support negative energy, the energy of combinations of EM fields is always positive. So - why have people even taken negative mass as seriously as they have?

PS: I did not win an FQXi essay prize this year, but fully four of my Facebook friends did. (Sabine won a couple times, is one too, which I appreciate.)

Neil,I don't know why you think this is an objection. It is perfectly possible to convert negative gravitational mass to negative gravitational energy. In fact one doesn't distinguish them to begin with, there's only the stress-energy tensor.

Sure, I know about the negative energy of the gravitational field. What I meant was: if you tried to make the NM ("negative mass") follow the analogy of: ordinary (positive mass) being converted into positive "light energy" as photons, also as electromagnetic fields. There is no way for EM fields to display negative energy, their fields are always positive energy. That is my specific point. (REM that EM fields are plus energy due to opposites attracting, grav. fields have negative energy (altho hard to tie into the radiative expression) due to like-"charges" attracting. Thanks for replying soon to my late comment. Also, you are so kind to Zephyr, it shows you are not oppressive and too defensive.

PS: Maybe I am a robot, some of those pictures are hard to make out ...

I am not talking about the energy of the gravitational field. I am talking about the gravitational energy of some matter or gauge field. The gravitational energy (mass) as opposed to the inertial energy (mass). Well, your analogy is simply wrong, as you have demonstrated yourself.

OK ... Then my own point was a reductio ab absurdem of idea that mass can be fully negative in all respects. So, I am not trying to contradict your more specific points about negative mass in the context of GR, but rather that: we cannot generalize "negative mass" to be a full analogy to positive mass in all respects. The reason is, because we can't convert it to its equivalent of negative energy as EM rays. IOW, E = mc^2 would not work for negative mass.

Here's a "cute" way to illustrate. Suppose I had some negative-mass U-235. Could I make a "negative A-bomb" out of it, that would freeze things with negative energy instead of burning them with heat? No. It would make a cool (;-) sci-fi story, but there is no way that light can have negative energy and absorb to make things colder. Maybe it was confusing to try and enlarge beyond your specific reference, but I wanted people to understand that "negative mass" does not work as a full counterpoint to positive mass in all respects. Best.